Boolean Circuit Problems
Problem 1. Adding It Up
How could we use boolean circuits to build a calculator? Let's investigate... a) Suppose that you have a one-bit number A, and a one-bit number B. What truth table represents the product of multiplying A times B? What circuit could you build to compute that truth table?
A B Product 0 0 0 1 1 0 1 1
Circuit: b) Suppose now that you want to add A and B instead of multiplying them. However, 1+1 is 2, which is bigger than either 0 or 1 – so now we need two circuits: one to compute the sum bit, and one to compute a carry bit. Fill in the rest of the truth table, and write the two circuits.
A B Sum Carry 0 0 0 1 1 0 1 0 1 1 0 1
Circuit for Sum:
Circuit for Carry: c) An “adder” circuit component, as used to add up numbers longer than just one bit, actually needs to add up three bits – A, B, and C. However, we still only have the sum bit and the carry bit as the two outputs. Write the two circuits for this truth table.
A B C Sum Carry 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1
Circuit for Sum:
Circuit for Carry: Problem 2. Functional Completeness
A set of logic gates is called functionally complete if you can use those logic gates to construct any other logic gate. a) {AND, OR, NOT} is a set of functionally complete logic gates. Demonstrate this by example: construct an XOR gate using only AND, OR, and NOT. Draw either a circuit or a formula for XOR in the space below.
b) NAND is functionally complete: all you need are NAND gates in order to make any other gate. Demonstrate this by constructing {AND, OR, NOT} from NAND gates. Draw either a circuit or a formula for each logic gate.
NOT
AND
OR
Assuming {AND, OR, NOT} is functionally complete, you've just proved that NAND is, also! Problem 3. No Inverters? from 6.045 Problem Set 1, Spring 2013
A Boolean circuit C is called monotone if changing any of the n input wires x1 … xn from 0 to 1 can only ever change the output C(x1...xn) from 0 to 1, never from 1 to 0. a) Argue that any circuit made of AND and OR gates must be monotone. (Discuss with partner) b) Make an example of a truth table for a monotone circuit which has three inputs. Write in the truth table below. (There are many possible correct answers.)
x1 x2 x3 C(x1,x2,x3) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
c) Show that your truth table can be constructed from AND and OR gates – no NOT gates needed! Draw the circuit that creates this truth table, below. You may use the constants 0 and 1 as “freebies” where necessary, as inputs instead of x1, x2, and x3.
d) Argue that any monotone truth table must be able to be made by a circuit of AND and OR gates. (Discuss with partner) Problem 4. Inverter Alchemy!
This problem is very challenging. If you decide to tackle it, you might want to download Logicly
Sometimes, people just expect the impossible. You were given only two inverters (aka NOT gates) to work with. But your assignment is to take in three inputs – A, B, and C – and produce their inversions: not A, not B, and not C.
Strange though it sounds, this task is actually possible! Prove it by figuring out what circuit will accomplish the task.